Hadron decays | Notion

First we can look at the interaction lifetimes of the different interactions

Interaction	Lifetime ($s$)
Strong	$10^{-22}-10^{-24}$
Electromagnetic	$10^{-16}-10^{-21}$
Weak	$10^{-7}-10^{-13}$

We define a rough approximation for these

$$ \frac{\alpha_\text{em}}{\alpha_\text{s}}\sim \sqrt{\frac{\tau_\text s}{\tau_\text{em}}} $$

🗒️ Note: for low mass hadron we have $\alpha_\text{s}\sim 1$ so the ratio of lifetime $\frac{\tau_\text{em}}{\tau_\text{s}}\sim 10^4$

We looked at strong compared to EM but we can see that weak decays are very supressed
- At low energy scales the virtual $W$ boson has an incredibly short lifespan due its mass making its production and thus the decay less likely
💃 Example: $n\to p+ e^- +\overline \nu_e$ energy produced is $\Delta E\sim 0.8 \,\rm MeV$ but the mass of the $W$ boson is $m_W\sim 80 \,\rm GeV$ making it obviously very supressed

🗒️ Note: for low mass hadrons we have $\frac{\alpha_\text{w}}{\alpha_\text{s}}\sim \sqrt{\frac{\tau_\text s}{\tau_ \text w}}$ which gives $\frac{\alpha_\text w}{\alpha_\text s} \sim 10^{-7}$

Conservation laws in hadron decays

Force	Conserved quantum number
Strong	$B,Q,S,C,\tilde B, J^{PC}$
Electromagnetic	$B,Q,S,C,\tilde B,L_e,L_\mu,L_\tau,J^{PC}$
Weak	$B,Q,L_e,L_\mu,L_z,J$

💼 Case: lets consider $K^-+p\to \pi^0 + X^0$ via strong interaction where $X$ is to be found

Looking at the quantum numbers we can write our interaction as follows

Quantum numbers	$K^-$	$p$	$\to$	$\pi^0$	$X^0$
$S$	$-1$	$0$	$\rightarrow$	$0$	$(-1)+(0)-(0)=-1$
$C$	$0$	$0$	$\rightarrow$	$0$	$(0)+(0)-(0)=0$
$\tilde B$	$0$	$0$	$\rightarrow$	$0$	$(0)+(0)-(0)=0$
$B$	$0$	$1$	$\rightarrow$	$0$	$(0)+(1)-(0)=1$
$Q$	$-1$	$+1$	$\rightarrow$	$0$	$(-1)+(+1)-(0)=0$

🗒️ Note: for the last column we used the conserved quantum numbers in the strong interaction

💎 Conclusion: $X^0$ is a neutral baryon with $uds$ constituents so it is a $\Lambda^0$

Quark diagrams

💃 Example: $\Pi^++p\to \Lambda^{++}\to \Pi^+ +p$

📜 Rules:

Time ordering goes from left to right

Quark arrows point right

Antiquark arrows point left

Continuity of quark lines reflects quark-antiquark production in pairs

🗒️ Note: we don’t write down the intermediate doubly charge $\Lambda ^{++}$ in the diagram

We can use these diagrams to check if certain decays are allowed 💃 Example: consider $K^-+p\to \Pi^0+\Lambda^0$

💎 Conclusion:

The interaction is allowed since all the quark lines follow the diagram rules
The intermediate particle is $uds$

🗒️ Note: here the intermediate particle $\Lambda^0$ is the only that respects conservation rules

Forbidden interaction

💼 Case: lets consider the $\pi^0\to3\gamma$